On the number of real classes in the finite projective linear and unitary groups Elena Amparo and C. Ryan Vinroot Abstract We show that for any n and q, the number of real conjugacy classes in PGL(n; Fq) is equal to the number of real conjugacy classes of GL(n; Fq) which are contained in SL(n; Fq), refining a result of Lehrer, and extending the result of Gill and Singh that this holds when n is odd or q is even. Further, we show that this quantity is equal to the number of real con- jugacy classes in PGU(n; Fq), and equal to the number of real conjugacy classes of U(n; Fq) which are contained in SU(n; Fq), refining results of Gow and Macdonald. We also give a generating function for this common quantity. 2010 Mathematics Subject Classification: 20G40, 20E45, 05A15 1 Introduction It was proved by G. I. Lehrer [6] that the number of conjugacy classes in the finite projective linear group PGL(n; Fq) is equal to the number of conjugacy classes of the finite general linear group GL(n; Fq) which are contained in the finite special linear group SL(n; Fq). I. G. Macdonald [7] showed that the number of conjugacy classes in the finite projective unitary group PGU(n; Fq) is equal to the number of conjugacy classes of the finite unitary group U(n; Fq) which are contained in the finite special unitary group SU(n; Fq) (although this number is different than the number of conjugacy classes in PGL(n; Fq) in general). Meanwhile, R. Gow [5] considered the number of real conjugacy classes in GL(n; Fq) and in U(n; Fq), where a conjugacy class of a finite group G is real if whenever g is in the class, then −1 so is g . In particular, Gow noted [5, pg. 181] that the number of real classes of GL(n; Fq) is equal to the number of real classes of U(n; Fq). More recently, N. Gill and A. Singh [3, 4] classified the real conjugacy classes of PGL(n; Fq) and SL(n; Fq). They noted [4, after Theorem 2.8] that when q is even or n is odd, the number of real classes of PGL(n; Fq) is equal to the number of real classes of GL(n; Fq) which are contained in SL(n; Fq). In this paper we prove that this equality holds for all n and q. Moreover, we show that this number is equal to the number of real classes of PGU(n; Fq), and is equal to the number of real classes of U(n; Fq) which are contained in SU(n; Fq). That is, we extend the result of Gill and Singh, and give a refinement of the results of Lehrer, Macdonald, and Gow. This paper is organized as follows. In Section 2, we establish notation for partitions and linear and unitary groups over finite fields, and we give an overview of the sets of polynomials over finite fields which we need. We finish this section with an important enumeration in Lemma 2.1. In Section 3, we describe the real conjugacy classes for the groups of interest. In particular, in Section 3.1 we give known results for the real classes in GL(n; Fq) and U(n; Fq), Gill and Singh's enumeration of the number of real classes of GL(n; Fq) which are contained in SL(n; Fq), and we explain why this is equal to the number of real classes of U(n; Fq) which are contained 1 in SU(n; Fq). In Section 3.2, we classify the real classes of PGU(n; Fq) by following the methods of Gill and Singh for PGL(n; Fq). In Lemma 3.8 we give an enumeration of the real classes of PGU(n; Fq), and show this is equal to the number of real classes of PGL(n; Fq). Finally, in Section 4 we prove our main result in Theorem 4.1, where the work left to be done is to prove that the number of real classes of PGL(n; Fq) is equal to the number of real classes of GL(n; Fq) which are contained in SL(n; Fq). We accomplish this by computing a generating function for each quantity, which has a particularly nice form. Acknowledgements. The second-named author was supported in part by a grant from the Simons Foundation, Award #280496. 2 Preliminaries For positive integers n; m, we denote their greatest common divisor by (n; m). We let jnj2 denote the largest power of 2 which divides n, or the 2-part of n. If G is a group with g 2 G, then jgj will denote the order of the element g, and jgj2 will denote the 2-part of the order of g. 2.1 Partitions Given an integer n ≥ 0, we denote a partition ν of n as ν = (1m1 2m2 3m3 ··· ); P such that i≥1 imi = n. Each integer mi = mi(ν) ≥ 0 is the multiplicity of the part i in ν. We can also denote the partition ν by ν = (ν1; ν2; : : : ; νl); Pl such that j=1 νj = n, and νj ≥ νj+1 ≥ 0 for j < l. Then we have mi(ν) is the number of j such that νj = i. We also assume each νj > 0 unless n = 0, in which the unique partition of 0 is considered the empty partition. We let Pn denote the collection of all partitions of n. 2.2 Linear and unitary groups over finite fields For any prime power q, we let Fq denote a finite field with q elements, and we fix an algebraic ¯ × ¯× closure Fq. We let Fq and Fq denote the multiplicative groups of nonzero elements in these fields. ¯ ¯ Let GL(n; Fq) denote the group of invertible n-by-n matrices over Fq, and we identify ¯ ¯× ¯ q GL(1; Fq) with Fq . Define the standard Frobenius map F on GL(n; Fq) by F ((aij)) = (aij), and so the fixed points of F give the general linear group over Fq: ¯ F GL(n; Fq) = GL(n; Fq): We let SL(n; Fq) denote the special linear group over Fq, or the elements of determinant 1 in × GL(n; Fq). The center of GL(n; Fq) is the group of scalar matrices, which is isomorphic to Fq . × We identify the scalar matrices with the group Fq by a slight abuse of notation. The projective linear group, which we denote by PGL(n; Fq), is the general linear group modulo its center: × PGL(n; Fq) = GL(n; Fq)=Fq : 2 ~ ¯ Define the map F on GL(n; Fq) by composing F with the inverse-transpose map, so for ¯ (aij) 2 GL(n; Fq), we have ~ > q −1 q −1 F ((aij)) = (aij) = (aji) : ~ We define the unitary group over Fq, which we denote by U(n; Fq), to be the group of F -fixed ¯ points in GL(n; Fq): ¯ F~ GL(n; Fq) = U(n; Fq): ~ Alternatively, one can define U(n; Fq) to be the group of F -fixed points in GL(n; Fq2 ), which n is also the isometry group of the Hermitian form on the vector space Fq2 defined by hv; wi = >vF (w), where v; w are viewed as coordinate vectors and F is the q-power map on coordinates. ¯× F~ × We identify U(1; Fq) with (Fq ) , which is the multiplicative subgroup of Fq2 of order q + 1. Denote this cyclic group by Cq+1. The special unitary group SU(n; Fq) is then defined as the group of determinant 1 elements in U(n; Fq). The center of U(n; Fq) is the group of scalar matrices with diagonal entries from Cq+1, and we again identify this group of scalars with Cq+1. The projective unitary group PGU(n; Fq) is the unitary group modulo its center, that is, PGU(n; Fq) = U(n; Fq)=Cq+1: When n = 0, we take each of the linear and unitary groups described above to be the group with one element. 2.3 Polynomials over finite fields In this section we define several sets of polynomials over finite fields which we need in order to describe conjugacy classes. Let t be an indeterminate, and for a finite field Fq we let Fq[t] denote the collection of polynomials in t with coefficients from Fq. We will primarily be interested in monic polynomials with nonzero constant term, and so we denote this collection of polynomials over Fq by Mq[t]. Given a polynomial f(t) 2 Mq[t] with deg(f(t)) = d, we define the reciprocal polynomial of f(t), denoted by f ∗(t), by f ∗(t) = f(0)−1tdf(t−1); d d−1 ∗ d −1 d−1 −1 −1 so if f(t) = t + ad−1t + ··· + a1t + a0, then f (t) = t + a1a0 t + ··· + ad−1a0 t + a0 .A ∗ ¯× polynomial f(t) 2 Mq[t] and its reciprocal f (t) have the relationship that α 2 Fq is a root of −1 ∗ f(t) if and only if α is a root of f (t) (with the same multiplicity). A polynomial f(t) 2 Mq[t] ∗ ¯× −1 is called self-reciprocal when f(t) = f (t), or when α 2 Fq is a root of f(t) if and only if α is a root with the same multiplicity. Note that the constant term is a0 = ±1 necessarily for a self-reciprocal polynomial. Now let rq;d denote the number of self-reciprocal polynomials in Mq[t] of degree d.